Weighted renormalized volume coefficients

Khaitan, Ayush

doi:10.48550/arxiv.2205.06018

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2023

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“…After this paper was finished, it was brought to our attention that many more relevant weighted invariants are constructed through the recent work of Khaitan [22,23]. In light of the results of the σ 2 -invariant and renormalized volume coefficients in closed Riemannian manifolds, we expect that results similar to ours also hold for the invariant constructed by Khaitan (cf.…”

Section: Introductionsupporting

confidence: 57%

Deformation of the Weighted Scalar Curvature

Ho,

Shin

2023

SIGMA

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Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547], we study in this paper the deformation of the weighted scalar curvature. By studying the kernel of the formal $L_\phi^2$-adjoint for the linearization of the weighted scalar curvature, we prove several geometric results. In particular, we define a weighted vacuum static space, and study locally conformally flat weighted vacuum static spaces. We then prove some stability results of the weighted scalar curvature on flat spaces. Finally, we consider the prescribed weighted scalar curvature problem on closed smooth metric measure spaces.

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Section: Introductionsupporting

confidence: 57%